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Creating a map from an arbitrary surface to a flat domain is the age old problem of cartography. In computer graphics applications this problem is often presented as the quest for low distortion mappings from triangle meshes to a flat domain. In my talk I will present a new approach to the construction of such mappings based on circle patterns. Here the triangle mesh is treated as a collection of circumcircles (one to each triangle) with intersection angles (one to each edge). This description admits a mathematically very clean formulation of the notion of discrete conformality with the desired mapping being the unique solution to a convex energy minimization in the unknown circle radii. Little more than a black box minimization software is required to implement this approach. The resulting mappings are of excellent quality in terms of angle distortion. But, as one would expect, they suffer from at times high area distortion. To ameliorate this shortcoming, as well as deal with arbitrary topology surfaces (without cutting them first!), we employ cone singularities. The mathematical theory of circle patterns accommodates these with no changes. The final result are global (seamless) discrete conformal texture maps from arbitrary topology surfaces with low area distortion.
Joint work with Liliya Kharevych and Boris Springborn.